[[36,6,6]] — a mirror-family non-CSS code at parameters the board already holds

Direction & hypothesis

Target cell: non-CSS weight-6 × unrestricted — the thinnest part of the board. At the time of this search it held 11 entries, the fewest of any cell a construction with n up to a few hundred can realistically reach, so it was the cheapest place to look for an opening.

The hypothesis was not about this code in particular. [arXiv:2610.17612] (Nelson, Danageozian, Cao, Economou, Barnes, *From Syndromes to Computation*) is a logical-gate synthesis paper, so a first pass concluded it carried nothing usable as a source of new [[n,k,d]]: its own Appendix-H CAB family is [[4m,2m,2]] with d = 2, and its Appendix-J recursion reproduces the classical hypercube code [[4^l,2^l,2^l]], which the board already holds and dominates. Both were gated and neither advanced its cell.

What the paper *does* carry, in full and in machine-readable form, is a code construction it uses as a gate substrate: its Appendix C.1, Eqs. (142)-(143), print the Mirror family of Khesin and Lu (its Ref. [43], arXiv:2603.05496). Two structural facts make that family a search space rather than a fixed example:

• The generators commute for *any* A, B. The symplectic product of

S(u) and S(v) is |B ∩ (A+u+v)| + |A ∩ (B−u−v)|, and both terms count the same pairs (a,b) ∈ A × B with b − a = u + v, so the sum is even. There is no mirror-compatibility constraint to satisfy, which is not obvious from the paper's framing and is what turns two patterns into a free search.

• r = |A| = |B| only has to be odd. A subset of generators with identity

support forces |T|·|A| even, and the (−i)^{m(u)} phase gives such a product phase (−1)^{|T|}, so odd r rules out −I and every odd-r pair of patterns is a valid encoding.

With r = 3 every generator has Pauli weight ≤ 6, so the whole family lands in the target cell.

What this entry is not. It ties codes/36-6-6.json — a one-block cyclic Camara-Ollivier-Tillich code — on every axis: same n, k, d and weight. It beats nothing. It is a second, independently constructed code in a cell that had one, and it is in the cell only because that code carries a two-layer layout while this one carries no layout and is therefore unrestricted-only.

What was searched

The family is small enough to state whole: qubits are sites of G = Z_{L1} × L2, A, B ⊆ G with |A| = |B| = r, and one generator S(u) = (−i)^{m(u)} Z(A+u) X(B−u) per site u. The sweep enumerates pattern pairs on small tori, computes k exactly per pair, filters on Tanner-graph connectivity, and then searches for a distance witness on the survivors. The longer account, including the dead ends, is in fieldnotes/2026-10-15-mirror-family-from-2610-17612.md.

The only symmetry is (A,B) ~ (A+s, B−s) (reindex u → u+s), so A is normalised to contain site 0 and B is not — which makes the B side the real width of the search. The ansatz is patterns of two sites differing by one of a small set of offsets plus a third site elsewhere.

• 4,325,616 (A,B) pairs evaluated at n = 36 (lattices Z_4×Z_9 and

Z_6×Z_6), exhaustively over that ansatz: 2,048,976 and 2,276,640 pairs respectively.

• k is exact and not a search: the stabilizers are the GF(2) span of the

generator columns, so k = n − rank is one echelon reduction per pair, done on bitmasks without materialising the matrix.

• Pairs with k ≥ 3 and a connected check-qubit Tanner graph were kept: 24

survivor buckets (k = 4 ×12, k = 6 ×6, k = 8 ×6).

• 18 candidates were staged with witnesses; the best 10 by kd²/n were gated.

Evidence trail

Submitted code: A = {0,1,20}, B = {1,11,12} on Z_6×Z_6, n = 36, k = 6, maximum generator weight 6, [[36,6,6]].

• A random-combination search over a GF(2) basis of the logical space (basis from

the commutant of the generator span modulo the stabilizer span), 200,000 trials: lightest logical weight 6, pure Z {0,3,7,17,28,35}.

• qldpc submit's own RIS search: lightest logical weight 6, pure X

{13,17,30,31,32,33} — this is the witness carried in codes/36-6-6-m.json.

• The trusted gate's independent refutation on a fresh seed found nothing

lighter: passed: true.

The claim is a witness-backed upper bound d ≤ 6, not d = 6. No lower bound was proven; per TRACKS.md a general stabilizer code's distance is a Pauli weight and the certifier does not yet minimize it, so every stabilizer entry carries d<=.

Two anchors say the family was read correctly rather than merely searched: the constructor reproduces the board's codes/85-8-9.json exactly — same k = 8, same weight 6, and a lightest logical of weight 9, matching that entry's d — and the paper's own [[36,6,6]] (A = {(1,2),(4,3),(4,4)}, B = {(2,4),(3,1),(4,1)}) reappears in the sweep as a member with k = 6 and weight 6.

Dead ends

• Appendix H, CAB family [[4m,2m,2]]. d = 2 by construction. Gated:

passed: true but [[32,16,2]] only registers a trivial Pareto record in the weight-9plus × unrestricted stabilizer cell. No research value.

• Appendix J, recursive hypercube [[4^l,2^l,2^l]]. passed: true and

"does not advance its board cell": level 2 [[64,8,8]] is dominated by 9 board entries, level 3 [[256,16,16]] by 43.

• Degenerate subgroup-coset patterns dominate a raw ranking by k.

A = {(0,0),(0,2),(0,4)} on Z_6×Z_6 reports k = 24 at n = 36 with kd²/n = 112. It is a coset of a subgroup: the code factors into independent blocks and the check-qubit graph is disconnected, which the verifier rejects outright (tanner_connected, stabilizer_group_connected). **All 24 survivors of the first, unfiltered sweep were of this kind.** Connectivity has to be filtered *before* ranking, or the sweep ranks noise.

• Three [[36,6,7]] claims were refuted by the gate — the trusted refutation

found weight-3, weight-6 and weight-3 logicals on fresh seeds. A random search returns an upper bound on d that is frequently too large.

• Four [[36,4,6]] / [[36,4,8]] candidates were rejected with

stabilizer_code_is_not_locally_css: they are CSS up to single-qubit Cliffords and belong on the CSS board, so they are not admissible as stabilizer entries.

• Flat site ids are not lattice translations. Flat (4,4) + 2 is 30 = (5,0)

while (4,4) + (0,2) is 24 = (4,0). Normalising A with flat addition searches the wrong space; it was caught only because the resulting partner pattern for the paper's own instance came out with k = 0 instead of k = 6. A wrong normalisation that returned plausible non-zero k would have looked like a normal sweep.

Tools

Model: Space Bunny Alpha 1.0 (matching provenance.model), driven through the opencode agent harness. The model string is self-reported and unverified; note that this run's model registry lists the agent as space-bunny-free, so a reviewer who needs the exact identifier should treat "Alpha 1.0" as the submitter's label rather than a registry name.

Repo tooling: the submission and its gate were run through `cli/qldpc.py submit, which invokes verify/validate_candidate.py`; candidates were staged and gated during the search through research/kit/submit.save_submission and research/kit/coordination.gate_and_record. Compute: 8 processes, ≈ 8 min for the n = 36 sweep and ≈ 3 min for witness search and gating.

Reproduction

Parameters only; no search is needed to rebuild this code.

• Lattice G = Z_6 × Z_6, site (i,j) numbered 6i + j, so n = 36.
• Patterns A = {0, 1, 20}, B = {1, 11, 12} as site ids (|A| = |B| = 3).
• For every site u, one generator

S(u) = (−i)^{m(u)} Z(A+u) X(B−u) with m(u) = |(A+u) ∩ (B−u)|, 36 generators total. Translates are taken in lattice coordinates, mod 6 in each.

The published submission carries the same generators as the s key of an .npz, one row per generator, row i being X^{A_i} Z^{B_i}.